// Numbas version: finer_feedback_settings
{"name": "Integration: By Substitution 3a ii (Negative and Fractional Powers)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: By Substitution 3a ii (Negative and Fractional Powers)", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $\\frac{nx^{n-1}}{x^n+a}$ using integration by substitution. $n$ is a positive or negative fraction.</p>", "licence": "None specified"}, "statement": "<p>Calculate \\[ \\simplify[fractionNumbers]{int(({n}x^{n/m-1})/({m}(x^{n/m}+{a})),x)}\\]</p>", "advice": "<p>Since this integral is of the form \\[ \\int g'(x)f(g(x))\\,dx,\\] we can use the method of substitution to calculate the solution.&nbsp;</p>\n<p>Firstly, we must make a change of variables from $x$ to $u$, where $u$ is equal to the 'inner' function $g(x)$.</p>\n<p>So, for \\[\\simplify[fractionNumbers]{int(({n}x^{n/m-1})/({m}(x^{n/m}+{a})),x)}\\]</p>\n<p>let $\\color{red}{u=\\simplify[fractionNumbers]{x^{n/m}+{a}}}.$</p>\n<p>Now, we need to calculate the differential, $du$, where \\[ du = \\left(\\frac{du}{dx}\\right)dx. \\]</p>\n<p>Differentiating $u$ with respect to $x$:</p>\n<p>\\[ \\frac{du}{dx}= \\simplify[fractionNumbers]{{n/m}x^{n/m-1}}.\\]</p>\n<p>Therefore, \\[ \\color{blue}{du = \\simplify[fractionNumbers]{{n/m}x^{n/m-1}}\\, dx}.\\]</p>\n<p>We can now rewrite the original integral in terms of $u$:</p>\n<p>\\[ \\int \\frac{\\color{blue}{\\simplify[fractionNumbers]{{n}x^{n/m-1}}}}{\\color{blue}{\\var{m}}(\\color{red}{\\simplify[fractionNumbers]{x^{n}+{a}}})}\\color{blue}{\\text{d}x} =&nbsp;\\int \\frac{1}{\\color{red}{u}}\\color{blue}{\\text{d}u}.\\]</p>\n<p><strong>(Note: It is important to see that both the function we are integrating, and the variable we are integrating with respect to, has changed.)</strong></p>\n<p>\\[ \\simplify[fractionNumbers]{int(1/u,u) = ln(abs(u)) + c}.\\]</p>\n<p>Finally, we must rewrite our solution back in terms of the original variable $x$:</p>\n<p>\\[ \\simplify[fractionNumbers]{ln(abs(u)) + c = ln(abs(x^{n/m}+{a}))&nbsp;+ c}.\\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(n/m)>1 and gcd(n,m)=1", "maxRuns": "100"}, "ungrouped_variables": ["a", "n", 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"useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "ln(x^({n/m})+{a})+c", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "answer": "ln(abs(x^({n/m})+{a}))+c", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, 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